What does it mean to cross load a carabiner? I know it's bad, and not to do it, but I don't really know what exactly it is.
A carabiner is designed to be loaded only along the long axis, near the spine (leftmost figure below). It will be weaker in any other direction of stress. Primary long-axis strength should be marked on the carabiner spine with an up-down arrow symbol, and is typically given in kilo-Newtons (one kN equals approximately 225 pounds of force).
Cross-loading is shown in the middle figure below. Strength in this orientation should be marked on the carabiner with a left-right arrow symbol.
The gate is a load-bearing member even when the carabiner is correctly loaded, therefore the "open gate" strength is much less, as should be marked with a symbol that looks like an open carabiner. Be aware that the inward gate strength is usually much less than the outward cross-load gate strength because only the locking sleeve holds the gate in place.
There is another weakness you need to be aware of: loading the carabiner at a point or direction away from the spine. This happens in "tri-axial" loading seen in the rightmost figure, or with a nose-hooked carabiner. This is or can be even weaker than the cross-loaded configuration.
Here is another instance of tri-axial loading (source). This is probably not as weak as a carabiner loaded the same way but upside down, because the offset-D has a shorter lever arm in this configuration, but it is still weak:
Likewise a carabiner loaded with multiple ropes, or a single rope with a bulky knot, even if loaded in the same and proper direction will be weaker on the strand(s) farther from the spine. (source)
It represents a two major things. I'm not going to illustrate, but the core of it is: